Question: The line passing through the points $ (2, 5) $ and $ (4, 13) $ intersects the y-axis at point $ (0, b) $. What is the value of $ b $? - Sterling Industries
The Line Through (2, 5) and (4, 13): How to Find Where It Meets the Y-Axis
The Line Through (2, 5) and (4, 13): How to Find Where It Meets the Y-Axis
Ever wondered how math helps decode patterns in data, design, or trends? A simple dot on a graph—say, (2, 5) and (4, 13)—holds quiet secrets about slopes, intercepts, and how lines shape visual communication. This common question matters not just to math students, but to anyone exploring how graphs inform decisions—from business analysts to designers. What is the exact point where this line crosses the y-axis, marked as (0, b)? Understanding this connection reveals more than a coordinate—it unlocks clarity in data interpretation, a core skill in Today’s information-driven landscape.
The Value of b: A Gateway to Line Equations
Understanding the Context
The value of $ b $ represents the y-intercept—the spot where the line crosses the vertical axis. For the line passing through (2, 5) and (4, 13), knowing $ b $ gives a complete picture of its slope and direction. This coordinate appears frequently in technical discussions, from data visualization to economic modeling. Discover how learning this concept improves your ability to analyze trends and make informed choices.
To find $ b $, begin by determining the line’s slope $ m $. With points $ (2, 5) $ and $ (4, 13) $, the slope is calculated as rise over run:
$m = \frac{13 - 5}{4 - 2} = \frac{8}{2} = 4$
With slope $ m = 4 $, using point-slope form from $ (2, 5) $:
$ y - 5 = 4(x - 2) $
Expanding gives:
$ y = 4x - 8 + 5 = 4x - 3 $
The y-intercept $ b $ appears when $ x = 0 $. Substituting:
$ y = 4(0) - 3 = -3 $
Thus, $ (0, -3) $, meaning $ b = -3 $.
Key Insights
Understanding the Line in Context
The slope of 4 indicates the line rises sharply—one unit right increases y by four. The negative y-intercept reflects this steep ascent on the graph, a detail valuable for diagnosing trends in data sets ranging from sales growth to user engagement. After scrolling through educational visuals, many notice this intersection point immediately highlights balance and growth patterns, reinforcing why slope-intercept form remains a cornerstone of analytical thinking.
Why This Question Is Resonating Now in the US Market
Curiosity about graphical relationships has sur
